+67 1. Solve each given equation and show your work. Tell whether each equation has one solution, an infinite number of solutions, or no solution. Explain your answers.
(a) 2x+4(x-1)=2+4x
(b) 25-x=15-(3x+10)
(c)4x=2x+2x+5(x-x)
a)
Solve for x:
4 (x-1)+2 x = 4 x+2
4 (x-1) = 4 x-4:
4 x-4+2 x = 4 x+2
Grouping like terms, 4 x+2 x-4 = (4 x+2 x)-4:
(4 x+2 x)-4 = 4 x+2
4 x+2 x = 6 x:
6 x-4 = 4 x+2
Subtract 4 x from both sides:
(6 x-4 x)-4 = (4 x-4 x)+2
6 x-4 x = 2 x:
2 x-4 = (4 x-4 x)+2
4 x-4 x = 0:
2 x-4 = 2
Add 4 to both sides:
2 x+(4-4) = 2+4
4-4 = 0:
2 x = 2+4
2+4 = 6:
2 x = 6
Divide both sides of 2 x = 6 by 2:
(2 x)/2 = 6/2
2/2 = 1:
x = 6/2
The gcd of 6 and 2 is 2, so 6/2 = (2×3)/(2×1) = 2/2×3 = 3:
Answer: |x = 3
b)
Solve for x:
25-x = 5-3 x
Add 3 x to both sides:
3 x-x+25 = (3 x-3 x)+5
3 x-3 x = 0:
3 x-x+25 = 5
3 x-x = 2 x:
2 x+25 = 5
Subtract 25 from both sides:
2 x+(25-25) = 5-25
25-25 = 0:
2 x = 5-25
5-25 = -20:
2 x = -20
Divide both sides of 2 x = -20 by 2:
(2 x)/2 = (-20)/2
2/2 = 1:
x = (-20)/2
The gcd of -20 and 2 is 2, so (-20)/2 = (2 (-10))/(2×1) = 2/2×-10 = -10:
Answer: |x = -10
c)
ANY VALUE OF X IS SOLUTION. BUT NO SPECIFIC SOLUTION.
